Diamond monotone: Difference between revisions

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A tuning for a [[rank]]-''r'' [[Prime limit|''p''-limit]] [[regular temperament]] is '''diamond monotone''', or '''diamond valid''', if it satisfies the following condition: the [[Odd-limit|''p''-odd limit]] [[tonality diamond]], when sorted by increasing size, is mapped to a tempered version which is also [[Wikipedia: Monotonic function|monotone]] increasing (i.e. nondecreasing).

In the original work by [[Andrew Milne]], [[Bill Sethares]] and [[James Plamondon]] — and to some extent on the wiki and in the regular temperament community — this tuning range was referred to simply as the "valid" tuning range.

In the original work by [[Andrew Milne]], [[Bill Sethares]] and [[James Plamondon]] – and to some extent on the wiki and in the regular temperament community — this tuning range was referred to simply as the "valid" tuning range<ref>Milne, A. J., Sethares, W. A., and Plamondon, J. (2007). Isomorphic controllers and Dynamic Tuning: Invariant fingering over a tuning continuum. Computer Music Journal, 31(4):15–32.</ref>.

The diamond monotone tuning range sets a boundary on any realistic possibility of correct recognition. Within this tuning range, the interval representing 6/5 will always be smaller than the interval representing 5/4 will be smaller than the interval representing 4/3. (As with the [[diamond tradeoff]] range, the precise boundary tunings depend on the intervals we wish to privilege - privileging those in p-limit tonality diamond is an arguably reasonable choice).

The diamond monotone tuning range sets a boundary on any realistic possibility of correct recognition. Within this tuning range, the interval representing 6/5 will always be smaller than the interval representing 5/4 will be smaller than the interval representing 4/3. (As with the [[diamond tradeoff]] range, the precise boundary tunings depend on the intervals we wish to privilege – privileging those in p-limit tonality diamond is an arguably reasonable choice).

The "empirical" range is likely to fall somewhere between diamond monotone and diamond tradeoff. Though, when one is using tempered spectra to match the tuning, it is possible the empirical range can be made wider.

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# The intervals between those are [6/5, 25/24, 16/15, 9/8, 16/15, 25/24, 6/5].

# We only care about the unique intervals, so we consider [6/5, 25/24, 16/15, 9/8].

# In vector form those are [{{monzo|1 1 -1}}, {{monzo|-3 -1 2}}, {{monzo|4 -1 -1}}, {{monzo|-3 2 0}}], respectively.

# If we map those using {{val| 1 ''a'' 5-2''a'' }} we obtain the tempered sizes [3''a'' - 4, 7 - 5''a'', ''a'' - 1, 2''a'' - 3].

# Now we need to make sure each of those are not negative, so we get a set of inequalities: ''a'' ≥ 4/3, ''a'' ≤ 7/5, ''a'' ≥ 1, ''a'' ≥ 3/2.

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 A tuning for a [[rank]]-''r'' [[Prime limit|''p''-limit]] [[regular temperament]] is '''diamond monotone''', or '''diamond valid''', if it satisfies the following condition: the [[Odd-limit|''p''-odd limit]] [[tonality diamond]], when sorted by increasing size, is mapped to a tempered version which is also [[Wikipedia: Monotonic function|monotone]] increasing (i.e. nondecreasing).
-In the original work by [[Andrew Milne]], [[Bill Sethares]] and [[James Plamondon]] — and to some extent on the wiki and in the regular temperament community — this tuning range was referred to simply as the "valid" tuning range.
+In the original work by [[Andrew Milne]], [[Bill Sethares]] and [[James Plamondon]] – and to some extent on the wiki and in the regular temperament community — this tuning range was referred to simply as the "valid" tuning range<ref>Milne, A. J., Sethares, W. A., and Plamondon, J. (2007). Isomorphic controllers and Dynamic Tuning: Invariant fingering over a tuning continuum. Computer Music Journal, 31(4):15–32.</ref>.
-The diamond monotone tuning range sets a boundary on any realistic possibility of correct recognition. Within this tuning range, the interval representing 6/5 will always be smaller than the interval representing 5/4 will be smaller than the interval representing 4/3. (As with the [[diamond tradeoff]] range, the precise boundary tunings depend on the intervals we wish to privilege - privileging those in p-limit tonality diamond is an arguably reasonable choice).
+The diamond monotone tuning range sets a boundary on any realistic possibility of correct recognition. Within this tuning range, the interval representing 6/5 will always be smaller than the interval representing 5/4 will be smaller than the interval representing 4/3. (As with the [[diamond tradeoff]] range, the precise boundary tunings depend on the intervals we wish to privilege – privileging those in p-limit tonality diamond is an arguably reasonable choice).
 The "empirical" range is likely to fall somewhere between diamond monotone and diamond tradeoff. Though, when one is using tempered spectra to match the tuning, it is possible the empirical range can be made wider.
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 # The intervals between those are [6/5, 25/24, 16/15, 9/8, 16/15, 25/24, 6/5].
 # We only care about the unique intervals, so we consider [6/5, 25/24, 16/15, 9/8].
 # In vector form those are [{{monzo|1 1 -1}}, {{monzo|-3 -1 2}}, {{monzo|4 -1 -1}}, {{monzo|-3 2 0}}], respectively.
 # If we map those using {{val| 1 ''a'' 5-2''a'' }} we obtain the tempered sizes [3''a'' - 4, 7 - 5''a'', ''a'' - 1, 2''a'' - 3].
 # Now we need to make sure each of those are not negative, so we get a set of inequalities: ''a'' ≥ 4/3, ''a'' ≤ 7/5, ''a'' ≥ 1, ''a'' ≥ 3/2.